Greetings Doron:
Concerning you recent Opinion 68, I have to say emphatically that
you are overreacting again! Clearly the new results of Fox are
very dramatic; however, a quick internet search turned up slides
of a recent talk by Daniel J. Kleitman and Jacob Fox where it
is pointed out that Paul Erdős and Shizuo Kakutani already proved
in 1943 that the NEGATION of the Continuum Hypothesis is equivalent
to the equation x1 + x2 - x3 - x4 = 0 being countably-regular (i.e.,
with respect to countable partitions of the reals). This is a
result of the same type as those of Fox (if not as refined and not
as dramatic -- as far as the linear equation is concerned), so
if you are to be shocked, shocked by such things, you should have
been in a state of shock for the last 60 years!
Remember the old results of Hausdorff-Banach-Tarski, which in
particular claim that a solid sphere can be "broken" into finitely
many subsets and "reassembled" as two spheres of the same size
as the original ball. This method can only be done with a well
ordering of the real numbers, and a well ordering as a subset of
the plane is always a Lebesgue non-measurable set. In fact,
Sierpinski showed many years ago that a non-principal ultrafilter
in the Boolean algebra of all sets of integers also implies the
existence of a non-measurable set.
The lesson here is that arbitrary sets of reals can be monstrously
horrible. So? We have known this for a very long time. And don't
forget that people once hated the idea that there could be a
continuous function nowhere differentiable. But they learned
to live with these things and pay more attention to the non-
pathological functions, sets, and spaces.
The area where you -- as a combinatorial personality -- should
really be worried concerns the recent work of Harvey Friedman.
He finds propositions of FINITE combinatorics which are "true"
but can only be proved with the aid of very large cardinals.
He is speaking of cardinal numbers so large they make the
aleph_n cardinals look like baby toys. You can say these
"theorems" are not "interesting", but I personally think you
would be wrong. Ask Harvey for a quick lesson about his results.
And best wishes for the New Year!
Prof. Dana S. Scott
1149 Shattuck Avenue
Berkeley, CA 94707-2609
------
Tel: (510) 527-5287